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Theorem lmhmplusg 15801
Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
lmhmplusg.p  |-  .+  =  ( +g  `  N )
Assertion
Ref Expression
lmhmplusg  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M LMHom  N
) )

Proof of Theorem lmhmplusg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2283 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2283 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 eqid 2283 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2283 . 2  |-  (Scalar `  N )  =  (Scalar `  N )
6 eqid 2283 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 15790 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantr 451 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 15789 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
109adantr 451 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 15787 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1211adantr 451 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  N
)  =  (Scalar `  M ) )
13 lmodabl 15672 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Abel )
1410, 13syl 15 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  Abel )
15 lmghm 15788 . . . 4  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
1615adantr 451 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
17 lmghm 15788 . . . 4  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
1817adantl 452 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  G  e.  ( M  GrpHom  N ) )
19 lmhmplusg.p . . . 4  |-  .+  =  ( +g  `  N )
2019ghmplusg 15138 . . 3  |-  ( ( N  e.  Abel  /\  F  e.  ( M  GrpHom  N )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M  GrpHom  N ) )
2114, 16, 18, 20syl3anc 1182 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M  GrpHom  N ) )
22 simpll 730 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  e.  ( M LMHom  N ) )
23 simprl 732 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  M ) ) )
24 simprr 733 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  y  e.  ( Base `  M
) )
254, 6, 1, 2, 3lmhmlin 15792 . . . . . 6  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x ( .s `  N
) ( F `  y ) ) )
2622, 23, 24, 25syl3anc 1182 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( F `  y
) ) )
27 simplr 731 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  e.  ( M LMHom  N ) )
284, 6, 1, 2, 3lmhmlin 15792 . . . . . 6  |-  ( ( G  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( G `  (
x ( .s `  M ) y ) )  =  ( x ( .s `  N
) ( G `  y ) ) )
2927, 23, 24, 28syl3anc 1182 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( G `  y
) ) )
3026, 29oveq12d 5876 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) )  =  ( ( x ( .s `  N
) ( F `  y ) )  .+  ( x ( .s
`  N ) ( G `  y ) ) ) )
319ad2antrr 706 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  N  e.  LMod )
3211fveq2d 5529 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
3332ad2antrr 706 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  (Scalar `  N
) )  =  (
Base `  (Scalar `  M
) ) )
3423, 33eleqtrrd 2360 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  N ) ) )
35 eqid 2283 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
361, 35lmhmf 15791 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
3736ad2antrr 706 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F : ( Base `  M
) --> ( Base `  N
) )
38 ffvelrn 5663 . . . . . 6  |-  ( ( F : ( Base `  M ) --> ( Base `  N )  /\  y  e.  ( Base `  M
) )  ->  ( F `  y )  e.  ( Base `  N
) )
3937, 24, 38syl2anc 642 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  y )  e.  ( Base `  N
) )
401, 35lmhmf 15791 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
4140ad2antlr 707 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G : ( Base `  M
) --> ( Base `  N
) )
42 ffvelrn 5663 . . . . . 6  |-  ( ( G : ( Base `  M ) --> ( Base `  N )  /\  y  e.  ( Base `  M
) )  ->  ( G `  y )  e.  ( Base `  N
) )
4341, 24, 42syl2anc 642 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  y )  e.  ( Base `  N
) )
44 eqid 2283 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
4535, 19, 5, 3, 44lmodvsdi 15650 . . . . 5  |-  ( ( N  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  N )
)  /\  ( F `  y )  e.  (
Base `  N )  /\  ( G `  y
)  e.  ( Base `  N ) ) )  ->  ( x ( .s `  N ) ( ( F `  y )  .+  ( G `  y )
) )  =  ( ( x ( .s
`  N ) ( F `  y ) )  .+  ( x ( .s `  N
) ( G `  y ) ) ) )
4631, 34, 39, 43, 45syl13anc 1184 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  N ) ( ( F `  y ) 
.+  ( G `  y ) ) )  =  ( ( x ( .s `  N
) ( F `  y ) )  .+  ( x ( .s
`  N ) ( G `  y ) ) ) )
4730, 46eqtr4d 2318 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) )  =  ( x ( .s `  N ) ( ( F `  y )  .+  ( G `  y )
) ) )
48 ffn 5389 . . . . 5  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  F  Fn  ( Base `  M )
)
4937, 48syl 15 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  Fn  ( Base `  M
) )
50 ffn 5389 . . . . 5  |-  ( G : ( Base `  M
) --> ( Base `  N
)  ->  G  Fn  ( Base `  M )
)
5141, 50syl 15 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  Fn  ( Base `  M
) )
52 fvex 5539 . . . . 5  |-  ( Base `  M )  e.  _V
5352a1i 10 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  M )  e. 
_V )
547ad2antrr 706 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  M  e.  LMod )
551, 4, 2, 6lmodvscl 15644 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
5654, 23, 24, 55syl3anc 1182 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  M ) y )  e.  ( Base `  M
) )
57 fnfvof 6090 . . . 4  |-  ( ( ( F  Fn  ( Base `  M )  /\  G  Fn  ( Base `  M ) )  /\  ( ( Base `  M
)  e.  _V  /\  ( x ( .s
`  M ) y )  e.  ( Base `  M ) ) )  ->  ( ( F  o F  .+  G
) `  ( x
( .s `  M
) y ) )  =  ( ( F `
 ( x ( .s `  M ) y ) )  .+  ( G `  ( x ( .s `  M
) y ) ) ) )
5849, 51, 53, 56, 57syl22anc 1183 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o F 
.+  G ) `  ( x ( .s
`  M ) y ) )  =  ( ( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) ) )
59 fnfvof 6090 . . . . 5  |-  ( ( ( F  Fn  ( Base `  M )  /\  G  Fn  ( Base `  M ) )  /\  ( ( Base `  M
)  e.  _V  /\  y  e.  ( Base `  M ) ) )  ->  ( ( F  o F  .+  G
) `  y )  =  ( ( F `
 y )  .+  ( G `  y ) ) )
6049, 51, 53, 24, 59syl22anc 1183 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o F 
.+  G ) `  y )  =  ( ( F `  y
)  .+  ( G `  y ) ) )
6160oveq2d 5874 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  N ) ( ( F  o F  .+  G ) `  y
) )  =  ( x ( .s `  N ) ( ( F `  y ) 
.+  ( G `  y ) ) ) )
6247, 58, 613eqtr4d 2325 . 2  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o F 
.+  G ) `  ( x ( .s
`  M ) y ) )  =  ( x ( .s `  N ) ( ( F  o F  .+  G ) `  y
) ) )
631, 2, 3, 4, 5, 6, 8, 10, 12, 21, 62islmhmd 15796 1  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M LMHom  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   Abelcabel 15090   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  nmhmplusg  18266  mendrng  27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ghm 14681  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lmhm 15779
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